The distortion in a lens is a function of the field height and stop location. Distortion is measured as the relative discrepancy between real image height and the value predicted by paraxial theory. In FIG. 1, the distance from the image center to the corner of the grid is indicated by h for the orthoscopic case (no distortion), and by h′ for the distorted grids. The relative distortion D is then given by (h′−h)/h.
Real-paraxial image height discrepancy varies with off-axis distance as h′−h=ah3+bh5+ . . . . Relative distortion varies as: D=(h′−h)/h=ah2+bh4+ . . . . The first coefficient, a, is positive for pincushion and negative for barrel distortion. Usually the quadratic term outweighs the higher-order terms and D essentially varies as h squared, increasing progressively toward the image corners.
The derivative of the distortion isdD/dh=2ah+4bh3+6ch5 